## Abstract

This paper classifies all memristors into three classes called *Ideal, Generic*, or *Extended* memristors. A subclass of Generic memristors is related to Ideal memristors via a one-to-one mathematical transformation, and is hence called Ideal Generic memristors. The concept of non-volatile memories is defined and clarified with illustrations. Several fundamental new concepts, including *Continuum-memory memristor, POP* (acronym for Power-Off Plot), *DC V-I Plot*, and *Quasi DC V-I Plot*, are rigorously defined and clarified with colorful illustrations. Among many colorful pictures the *shoelace* DC V-I Plot stands out as both stunning and illustrative. Even more impressive is that this bizarre *shoelace* plot has an *exact analytical* representation via 2 *explicit* functions of the state variable, derived by a novel *parametric approach* invented by the author.

### Keywords

- Memristor
- Continuum-memory memristor
- POP
- Power-Off plot
- DC V-I plot
- Quasi DC V-I plot
- Shoelace V-I plot
- Parametric approach
- Graphical composition
- Piecewise-Linear function (PWL)

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## Notes

- 1.
Our axiomatic definition requires a

*pinched**hysteresis loop*to be measured not only for one input signal, but for*all*possible periodic input signals with zero mean. In practice, only a finite number of measurements could be made. Our definition did*not*require*the same*pinched hysteresis loop to be measured whenever the same input signal is applied, because for*non*-*volatile*memristors, the pinched hysteresis loop depends not only on the input waveform*i*(*t*) or*v*(*t*), but also on the initial conditions of the relevant state variables, such as Fig. 42 of [5], where two*different*pinched hysteresis loops are measured for the same input current source*i*(*t*) = 10 sin*ωt*, but different initial states*x*(0) = −6.3 and*x*(0) = 6.3, respectively. - 2.
Readers puzzled by the dramatic difference between the 2 pinched hysteresis loops in Fig. 4b, d calculated from

*the same*memristor constitutive relation \( q = \hat{q}(\varphi ) \), and the same input*v*(*t*) = 1.2 sin*t*are referred to the exact calculations and graphical illustrations in Figs. 28 and 27 of [5], respectively. - 3.
- 4.
Although one can choose any piecewise-differentiable function \( x = \hat{x}(\varphi ) \), it is rare that its inverse function \( \varphi = \hat{x}^{ - 1} (x) \) has an analytical equation. We opted for a PWL function because not only it has an explicit equation, as shown in Table 6 of Appendix , its

*inverse*function is also a PWL function, and hence will also have an explicit equation. - 5.
Recall the 1:1 function we used to derive the memristor sibling in Fig. 10 is defined by \( x = \hat{x}\,(\varphi ) = 2.125\,\varphi - 1.875\left| \varphi \right| \).

- 6.
The same algorithm applies for voltage-controlled memristors .

- 7.
A current-controlled (resp., voltage-controlled) memristors is

*passive if*,*and only if*, its memristence*R*(*x*,*i*) ≥ 0 (resp. memductance*G*(*x*,*v*) ≥ 0). A memristor is*active if*,*and only if*, it is*not passive*. - 8.
Fig. 19d illustrates why the

*Coincident Zero*-*Crossing signature*is more general than that of a*pinched hysteresis loop fingerprint*; namely, since both*i*(*t*) and*v*(*t*) waveforms are*not*periodic, their associated Lissajous figure is*not*a*closed*loop, but an unending loci, which if left to continue printing unstopped would eventually lead to a uniformly blue color inside each lobe. But remarkably, both blue lobes would share a common*pinched*point at the origin at all times, thereby confirming the nonlinear device defined in Fig. 19a is a*memristor*. - 9.
The

*phase shift*between two periodic waveforms*i*(*t*) and*v*(*t*) of the same frequency is the distance in time measured between their two closest zero crossings. - 10.
The adjective

*asymptotically*is used in a*mathematical*sense meaning that*x(t)*will not arrive at*x*(*t*) =*x*(*Q*) at a*finite*time. - 11.
To simplify arithmetic, we pick

*G*_{0}= 1. In practice,*G*_{0}is a scaling constant chosen to fit the intrinsic memductance scale of the memristor. - 12.
An equilibrium point

*x(Q)*of a differential equation \( \dot{\varvec{x}} = \varvec{f}(\varvec{x}) \) is said to be*asymptotically stable*if a small ball placed initially at*x*(*Q*) will always return to*x*(*Q*) when it is displaced by an arbitrarily small perturbation of arbitrarily short duration by following the direction of motion indicated by the arrowhead along the dynamic route where the perturbed state \( \hat{x} = x(Q) + \Delta x \) is located. If the perturbed state \( \hat{x} \) did not return to*x*(*Q*), but remains motionless after the perturbation \( \Delta x \) became zero, then the equilibrium point is said to be*stable*. - 13.
- 14.
We choose the capital letters

*V*and*I*, instead of the conventional lower case letters*v*and*i*to distinguish them from the*H F v*-*i curve*(acronym for high-frequency*v*-*i*curve) exhibited by*all*Extended memristors when connected to high-frequency periodic signals. - 15.
Commercial simulators, such as various versions of SPICE , are even less reliable because the numerical algorithm it uses is incapable of solving complicated nonlinear equations.

- 16.
More accurate solutions can be found by using the graphically derived solutions as

*initial*guess for numerical softwares. - 17.
- 18.
In a future paper, we will design an oscillator by connecting a 7-volt battery (with the positive terminal connected to ground) in series with a positive inductor and the memristor defined by (16a–c). This battery will give rise to an unstable equilibrium point located at (−7, −63), thereby spawning a stable limit cycle via a super-critical Hopf Bifurcation mechanism [25].

- 19.
It is possible, however, to design an elaborate experimental set-up to observe the unstable green branch in Fig. 30.

- 20.
Any state equation \( \frac{dx}{dt} = g(x,v) \) where \( g\,(x,0) = 0 \), −∞ <

*x*< ∞ has a continuum memory consisting of all points of the*x*-axis. - 21.
Equation (51a) defines a passive memristor because its instantaneous power \( p(t) = i(t)v(t) = x^{2} (t)v^{2} (t) \ge 0 \) for any v(t) and for all times t.

- 22.
It is possible to build a nullator using an op-amp. Indeed, if one connects a resistor from the negative op-amp input terminal of an op-amp to its output terminal, then 2 op-amp input terminals becomes a virtual short circuit, where

*v*= 0 and*i*= 0! - 23.
An ordinary differential equation \( \frac{{d\varvec{x}}}{dt} = {\text{f}}(\varvec{x},t) \) is said to be non-autonomous when the time variable t appear explicitly in the equation, such as the case when the memristor is driven by non-constant voltage source.

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## Acknowledgements

The author wishes to thank Prof. Hyongsuk Kim, Zubaer Ibna Mannan, and Cheol Choi for their wonderful assistance in the production of this paper. He would also like to thank Dr. R. Stanley Williams from hp for detecting several errors. The author would like to acknowledge financial support from the USA Air force office of Scientific Research under Grant number FA9550-13-1-0136 and from the European Commission Marie Curie Fellowship, and the EU COST Action IC 1401.

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## Appendix

### Appendix

Since nonlinear algebraic, or differential equations in general has no analytical solutions, they are usually solved by numerical methods, via standard softwares, or circuit simulators
, such as SPICE
. Unfortunately, numerical softwares are not foolproof, and cannot find all solutions if the equation has more than one solutions. Piecewise-linear (PWL) methods are the best tools in such situations. In order to apply PWL methods, it is often desirable to represent a PWL curve by a PWL equation whose only nonlinearities are the *absolute-value* function \( y = \left| x \right| \), and the *signum function* *sgn* (*x*), defined in Table 7.

The good news is that unlike other nonlinear basis functions, the coefficients associated with the PWL formula presented in the following Table 6, all coefficients needed to specify any continuous PWL function can be obtained by inspection of the PWL curve! Simply label the the segment number consecutively from left to right, as segment 0, 1, 2,…, *n*, for an (*n* + 1)-segment PWL curve, with corresponding slope *m*_{0}, *m*_{1},…, *m*_{n}. Label the x-coordinate of each corresponding breakpoint as *X*_{1}, *X*_{2},…, *X*_{n}. Any continuous PWL curve has a unique PWL formula (shown in Table 6), with 2 coefficients *a*_{0}, *a*_{1}; *n* coefficients *X*_{1}, *X*_{2},…, *X*_{n}, and *n* coefficients *b*_{1}, *b*_{2},…,*b*_{n}.

The coefficient X_{j} is equal to the x-coordinate of breakpoint *j*. The coefficient a_{1} is equal to half the sum of the slope *m*_{0} and *m*_{n} of the leftmost segment 0 and the rightmost segment *n*, respectively.

The coefficient *b*_{j} is equal to half the difference between the slope *m*_{j} and *m*_{(j-1)} of segment *j* and segment (* j*−1), respectively. The coefficient *a*_{0} is chosen such that *y* is equal to the vertical intercept *f*(0) of the PWL curve when *x* = 0 (see Table 6).

All of these coefficients can be reconstructed from the following mnemonic rule: *Half Sum Half Difference then Null*.

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Chua, L. (2019). Everything You Wish to Know About Memristors but Are Afraid to Ask. In: Chua, L., Sirakoulis, G., Adamatzky, A. (eds) Handbook of Memristor Networks. Springer, Cham. https://doi.org/10.1007/978-3-319-76375-0_3

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